Diffusion of spin bearing molecules in porous media observably affects the nuclear magnetic resonance (NMR) signal. Inferring microstructural features of the pore from the diffusion NMR signal attenuation has proven to be of paramount value in a variety of applications from oil-well logging and dynamics of polymers to the diagnosis and monitoring of many diseases in the human body. The most commonly used NMR method with which to observe diffusion in porous media employs the pulsed field gradient (PFG) experiments in which a pair of pulsed magnetic field gradients is applied to encode displacements between the application of these two pulses.
Although the PFG experiments have been useful in characterizing pore microstructure, many additional features, particularly those related to different length scales of porous media, can be gleaned if different pulse sequences are employed. One such alternative is the multi-PFG experiment, which involves the application of repeated pairs of diffusion gradients. Variants of this pulse sequence have been considered and found useful in various applications. The simplest version of such sequences employs only two pairs of gradients; a spin-echo version of this double-PFG sequence is shown in FIG. 1A. In FIG. 1A, each pulsed-gradient spin-echo (PGSE) block, comprising a pair of diffusion gradients of duration δ, sensitizes the signal to motion that occurs during an interval Δ. The movements of molecules during the two encoding intervals are correlated when the mixing time tm is finite. G1 and G2 denote the diffusion gradients of the first and second encoding blocks, respectively. FIG. 1B shows another double-PGSE sequence that results from the simultaneous application of the second and third gradients of the sequence in FIG. 1A.
The acquisition and analysis schemes for double-PFG data depend on the structure to be examined. For example, the strength of the first and second gradients can be independently varied. When the diffusion process can be characterized locally by a diffusion tensor, then a two-dimensional Laplace transform can be employed to generate maps of diffusion coefficients depicting the correlations of motion during the two encoding periods. This approach has been applied to plant tissue as well as various phases of liquid crystals.
The double-PFG experiments have received increasing attention recently due to the realization that such experiments are sensitive to restricted diffusion even at diffusion wavelengths that are long compared to the pore dimensions. As used herein, diffusion wavelength is defined as the quantity Λ=(γδG)−1, wherein γ denotes the gyromagnetic ratio of the spins and G is the gradient magnitude. The long diffusion wavelength regime (Λ2>>a2, where a is a characteristic pore size) is sometimes referred to as the small-q regime, (2πqa)2<<1, where q denotes the wave number, defined through the relationship q=1/(2πΛ)=γδG/(2π).
The sensitivity of the double-PFG experiments to restricted diffusion in this regime is a very desirable property, which makes it possible to probe small pores using relatively small diffusion gradient strengths. Recent findings suggest that the dependence of the signal intensity on the angle between the two gradients, G1 and G2, may make it possible to determine the sizes of biological cells using moderate gradient strengths. Although such an angular dependence was predicted by P. P. Mitra, Phys. Rev. B 51:15074 (1995), Mitra considered only special limiting cases of the double-PFG experiment. (|G1|=|G2|, Δ→∞, δ=0 and tm=0 or tm→∞), which are difficult to achieve in practice. Moreover, when even one of these conditions is not fully met, systematic errors in the estimations of the microstructural features are unavoidable.
The observation of diffusion of spin-labeled molecules provides an indirect means to probe geometries whose characteristic dimensions are smaller than the voxel resolution of conventional noninvasive MR imaging techniques. Incorporation of pulsed field gradients in MR pulse sequences has made it possible to conveniently measure the diffusion characteristics of the sample. One striking observation was that in materials with an ordered structure, signal attenuation (when plotted as a function of q=γδG/2π, where γ is the gyromagnetic ratio, δ is the diffusion pulse duration and G is the gradient vector) exhibited non-monotonic behavior. Specifically, when the wave-number (q=|q|) assumed certain values depending on the spacing between the restrictions, there was an almost perfect phase cancellation resulting in very small signal values. This fact was exploited to determine the compartment size. Perhaps the simplest system that exhibits this behavior is diffusing molecules sandwiched between two infinite parallel planes separated by a distance L. For this geometry, the diffraction dips occur when the wave-vector takes the values q=n/L, where n=1, 2, 3, . . . is the index of the diffraction well.
Another observation regarding the diffraction patterns is that in anisotropic samples, if diffusion is almost free along certain directions but restricted along others, the diffraction pattern is very sensitive to the direction of the diffusion gradient; It is observed only when the diffusion gradients are almost perpendicular to the restricting walls. This fact can be exploited to estimate fiber orientations.
Despite its potential, the application of diffraction patterns has been limited mostly because of the demanding nature of the measurements. In particular, the q-value has been required to exceed the reciprocal of the spacing between restricting barriers. This can be achieved by increasing the magnitude of the diffusion gradients or their durations. However, hardware limitations prohibit increasing the gradient strength beyond a certain point. Although increasing the gradient duration is possible, the violation of the narrow pulse approximation pushes the diffraction dips towards even larger q-values which in turn makes the pore size estimations less accurate. Another characteristic of the diffraction patterns that limits their widespread use is that the diffraction wells are observable when the diffusion time is long. In short diffusion times, the signal attenuation curve is quite featureless. In contrast to the requirement on q described above, this makes it difficult to measure the sizes of larger pores.
Most porous materials of interest are composed of pores with a broad distribution of sizes. In this case the diffraction wells are not observable at all, and the estimation of an average pore size may be possible using sophisticated methods but not directly from the locations of the diffraction dips. This problem is especially important for the estimation of cell sizes in biological specimens because of their large variability. Consequently, the non-monotonicity of the MR signal has been observed only in the very coherently organized regions of biological tissue such as corpus-callosum of fixed rat brains.
Some PFG-based measurements have involved the inversion of the magnetization via the application of a series of subsequent 180° radio-frequency (RF) pulses and application of separate pairs of diffusion gradients before and after each of the RF pulses as shown in FIG. 1C, In FIG. 1C, tm denotes the waiting time between two consecutive pulsed-gradient spin-echo (PGSE) blocks. Typical applications of such pulse sequences are based on ‘idealized’ experimental conditions such as δ=0, Δ→∞ (wherein Δ is the diffusion pulse separation) and tm=0 or tm→∞. Such conditions are difficult or impossible to achieve in many practical applications.